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3
votes
1answer
111 views

Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
0
votes
0answers
11 views

Schaten p norm of block matrices

Let $A=D\oplus 0$ be a diagonal Hermitian matrix and $B$ is an invertible Hermitian matrix with $(1,1)$ block being $B_{11}$ and $B_{11}$ and $D$ have the same dimensions. Then is it true that if $(1+|...
2
votes
1answer
137 views
+50

Relation between coefficients of expansions

Related to Relations between coefficients of expansions of a rational function at 0 and infinity I commented at the linked question that the question seemed less about what happened "at infinity", ...
0
votes
0answers
5 views

Radon transform range theorem and radial functions

In dimension 2, the Radon transform range theorem states that a function $g(t,\theta)$ can be represented as a Radon transform of some function $f(x,y)$ (i.e. $g=R[f]$) if and only if for all integers ...
0
votes
0answers
20 views

Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...
6
votes
1answer
363 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
4
votes
1answer
165 views

Conjectured primality test for specific class of $N=k \cdot 6^n+1$

Can you provide a proof or a counterexample for the claim given below? Inspired by Theorem 5 in this paper I have formulated the following claim: Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\...
1
vote
1answer
51 views

Sampling i.i.d. variables with restrictions

General Problem: Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "...
4
votes
0answers
54 views

Automorphisms of $\mathbb{C}[x, y, z]$ over $\mathbb C[x]$

What are the automorphisms of $\mathbb{C}[x, y, z]$ fixing $\mathbb{C}[x]$? I.e. those automorphisms $\phi:\mathbb{C}[x, y, z]\to\mathbb{C}[x, y, z]$ s.t. $\phi(x) = x$. I am interested in complete ...
2
votes
0answers
26 views

Real-world example of a Banach *-algebra with a nonzero *-radical

Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
2
votes
0answers
33 views

Sections of non-reduced schemes

Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...
0
votes
1answer
25 views

Distribution of the direction of Gaussian random variable

Let $X$ be a complex normal random variable. (Or, equivalently, a 2D real normal.) Is it possible to say anything useful about the distribution of the phase of $X$? Is it possible to do estimation on ...
2
votes
1answer
78 views

On a particular proof of “if the sharp of every real exists and every club contains a club constructible from a real, then $\delta^1_2 = \omega_2$”

I am referring to the proof of (4) implies (1) in Theorem 3.16 of Woodin's The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. His proof leverages on the fact that if the sharp of ...
1
vote
0answers
9 views

How do solutions to this quadratic congruence distribute as the number of factors grows?

This is a question that arose during a conversation with a colleague regarding Landau's fourth problem, which asks whether there are infinitely many primes of the form $n^2+1$. The Conjectured ...
6
votes
1answer
198 views
+50

Boundedness of total current in electrical network (Banded graph)

Consider the following symmetric matrix (adjacency matrix): $$A=(a_{ij})_{1\leq i,j\leq n}$$ such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
4
votes
1answer
308 views

Which books should I read in order to be prepared to study information geometry?

At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
6
votes
2answers
276 views

Curvature of nonsymmetric metric tensors?

Consider a smooth manifold $M$ of arbitrary dimension. We have notions of psuedo-Riemannian or Riemannian metrics on a manifold, and they differ in the slightest way of being positive-definite or not. ...
2
votes
0answers
33 views

Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
2
votes
0answers
76 views

Size issues (small/large categories) when defining stacks in the Algebraic/differentiable/topological setting

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note: We will not distinguish between ...
7
votes
1answer
220 views

Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor

It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$. Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
-2
votes
0answers
80 views

Asymptotically vanishing probabilities

Let $P_n$ be a sequence of probability measures on $(\mathbb{R}^{n},\mathcal{R}^{n})$, where $\mathcal{R}^{n}$ is the Borel $\sigma$-field associated to $\mathbb{R}^n$. For any sequence of measurable ...
-2
votes
1answer
164 views

Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

This is a cross-post of this MSE post that users commented that it is appropriate for MO. I want to know Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological ...
-2
votes
0answers
38 views

About the Notation $\int |d\mu|$ [migrated]

I see the notation $\int |d\mu|$ lots of times and I could not find the definition of this notation. I can guess what it means, but I want to know the mathematically precise meaning. Thank you!
0
votes
1answer
29 views

How to combine global standard deviation given several sample statistics?

Is there any approximation formula (best guess?) to calculate global std given multiple set statistics (size, mean, std)? I have an aggregated statistics from several sets. ...
0
votes
1answer
255 views

Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
7
votes
0answers
115 views

Can a covering space of the $p$-adic disc split over the circle?

Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$. Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...
1
vote
0answers
21 views

Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that $\{b_1,\ldots,b_n \}$ is a ...
1
vote
0answers
65 views

G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state. Consider the covariant GNS ...
2
votes
0answers
130 views
+50

The first part of the Hilbert sixteenth problem for elliptic polynomials

A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$. Inspired by the first part of the Hilbert ...
0
votes
1answer
31 views

An alternative description of normalized cochains in terms of tensor powers of the augmented ideal

I want to know if the following alternative of the normalized non-homogeneous cochains is already know. Let $G$ be a group and let ${\mathcal I}={\mathcal I}_G$ be its augmentation ideal, ${\mathcal ...
3
votes
3answers
302 views

Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$

I have two questions. $\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
-3
votes
0answers
68 views

$x^4=−1 \pmod p$ implies $p = 1 \pmod 8$ [closed]

Let $p$ be an odd prime. Show that $$x^4=−1 \pmod p$$ has a solution if and only if $⇔$ $p=1 \pmod 8$ Can someone help me with this, please? I really want to understand the steps.
-1
votes
0answers
79 views

The space of harmonic functions on an open set is infinite dimensional? [closed]

I want to prove that he space of harmonic functions on an open set $\Omega \subset \mathbb{R}^n $ , with $n \geq 2$, is uncountablely infinite-dimensional. I guess that I have to find a linearly ...
1
vote
0answers
21 views

Number of solutions to linear diophantine equations, with natural coefficients in a box

Let c, k, d $ \in \mathbb{N} $, let a, x $ \in \mathbb{N}^k $ suppose for all i $ \leq $ k, $ x_i \leq d $, $ a_i \in \mathcal{O}(d2^i) $ and $ \sum{a_ix_i} = c $ my question is for the value of c ...
4
votes
0answers
42 views

What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)? In even dimensions, all facets of the dual are ...
7
votes
0answers
147 views

Checking if a multiplication table represents a group [duplicate]

Given an $n \times n$ multiplication table, can one check if it represents a group in $o(n^3)$ time? All properties can be checked by mindless try-all possibilities loops: Whether there is an ...
1
vote
2answers
313 views

Two-dimensional Perron formula

There is a well-known Perron formula, which connects a mean value of certain arithmetic function with its Dirichlet series: $$ \sum_{n\le x} f(n) = {1\over 2\pi i} \int_{c-i\infty}^{c+i\infty} F(s) x^...
0
votes
0answers
327 views

Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$

Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
4
votes
1answer
70 views

Rates of convergence to Tracy-Widom?

$\renewcommand{\!}{\mathbf} \renewcommand{\Ai}{\operatorname{Ai}}$ One can define the Tracy-Widom distribution as the Fredholm determinant $F_2(t)=\det(\mathbf I-\mathbf A)$ where $$\mathbf A(x, y)=\...
4
votes
1answer
126 views
+50

Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...
1
vote
0answers
88 views

Varieties with everywhere good reduction isomorphic over every completion

Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the localization of $R$ at $\mathfrak{m}$ ...
1
vote
0answers
66 views

Surjectivity of multiplicative map

Let $S$ be a smooth complex algebraic surface, and $\mathcal{F}$ be a coherent sheaf on $S$. I want to consider $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{F} \boxtimes \mathcal{F}...
4
votes
0answers
104 views
+50

Batalin-Vilkovisky integral is invariant under infinitesimal deformation

This is the basic theorem of Batalin-Vilkovisky integral, and these are three main versions of the theorem. Articles about BV formalism almost certainly cite one of them, especially Schwarz's. However,...
0
votes
0answers
31 views

How prove this combinatorial-identities

if $n\ge 1$ be postive integers,and $x,y,z$ be any real numbers,such $x+y+z=n-1$,for any real number $a$.show that $$\dfrac{(-4)^n}{\binom{2x}{n}}\sum_{r+s=n,r,s\in Z}\dfrac{\binom{y}{r}\binom{y-a}{r}...
2
votes
1answer
94 views

Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
19
votes
5answers
695 views

Computation of fraction field of formal series over the integers

What is the fraction field $K$ of the domain $\mathbb Z[[X]]$? It is strictly smaller than the field of Laurent series $L=\operatorname {Frac}\mathbb Q[[X]]$, since $\sum_{i\geq 0}\frac {X^i}{i!}\in ...
2
votes
0answers
24 views

Multiplication formula in Grassmannian cluster categories

Grassmannian cluster categories are studied in A categorification of Grassmannian cluster algebras and Cluster categories from Grassmannians and root combinatorics. The category $CM(B_{k,n})$ of Cohen-...
2
votes
2answers
73 views

Finding the nearest quadratic Bézier curve

Given a set of three-dimensional quadratic Bézier curves. I'm looking for some analytical solution to find the nearest curve to an arbitrary point in space. Example I already have a brute force ...
4
votes
0answers
88 views

The Jacobson radical as a bimodule

Let $A$ be a finite dimensional algebra with Jacobson radical $J$. Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...
4
votes
1answer
344 views

An open problem on polynomials

Let $P(z)$ be a polynomial of degree $n$ with $|P(z)|\leq 1$ on $|z|=1$ and $P_m(z)$ be a partial sum of $P(z).$ How large $P_m(z)$ can be on $|z|=1?$ It is an open problem and I did not find any ...

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